Why did the Greeks not adapt/improve the phalanx a la the Romans' maniple?

This question is quite unfair on the Greeks for two reasons. One, their lack of an organised military made it very difficult for them to change the way they fought. Two, they absolutely did improve their phalanx and their battle tactics over time. It would be very wrong to think of Greek heavy infantry tactics as stagnant for centuries - but it's important to contextualise when and where changes happened.

To start with the first point: no Greek city-state had an army. They simply never created anything we would recognise as a professional military, with state pay for professional soldiers, logistical support, facilities, training schools, officer academies, etc. What they had was a militia of citizens called up at need, armed with the weapons they could afford. At best, these ad-hoc armies were supplemented by a few standing units or mercenary contingents (most of which were also temporary, since Greek city-states lacked the financial means to keep such things at their disposal permanently). Their lack of a professional army also prevented the development of a proper military mindset; Greek citizens did not like the idea of being ordered around, and only the Spartans ever seem to have learned to respect the concept of military authority as such.

The result of this is that a Greek city-state couldn't simply ordain that the citizens were henceforth to fight in a particular war. They did not supply their militia with weapons, and therefore relied on whatever people turned up with. Anyone who tried to introduce a new weapon or tactic had to go up against a citizen body protesting that there was nothing wrong with how they'd always done it, that the new way of doing things was just a lot of hard work, etc. In addition, they did not provide their citizens with compensation for time spent training, which meant in practice that they did not train. How much could the effectiveness of an untrained force be improved without any additional training?

The Greek way of war therefore changed very slowly. New ideas were constantly being tried by innovative commanders, but less inspiring leaders could not replicate them; new inventions were being made, but the means or the willingness to adopt them were lacking. For the most part, tactics remained very simple, because an untrained mob couldn't be expected to perform well in more complex scenarios. Most Greek city-states seem to have accepted the fact that a large semi-organised mass of heavy infantry, supplemented by cavalry and light troops, was the most effective form that their untrained citizen levy could take. Several commanders actually found out to their cost that it was a bad idea to try to make the proud free men of Greece work harder at being better soldiers.

Now, to the second point: the Greeks absolutely did improve the phalanx and its use in battle. Obviously, they originally developed this regular rank-and-file formation out of earlier disorganised masses of heavy infantry. There would have been no phalanx without this early effort to create a more serviceable heavy infantry formation. But its development continued. The Spartans famously introduced formation drill at some point in the 5th century BC, allowing them to maintain their formation while moving, and allowing them to perform rudimentary manoeuvres in battle. While this required more training than other Greeks were willing to submit to, other city-states developed the concept of picked troops - small detachments of strong, fit hoplites that could be relied upon to carry out specific tactical missions. In the course of the Classical period, some of these picked troops became standing forces, transforming into the elite core around which the hoplite militia could be built when the call came.

Throughout this period, too, more and more Greeks followed the Spartan lead in introducing uniform battle dress for maximum intimidation. The panoply was constantly being lightened to allow hoplites to operate more flexibly and to increase their endurance. The Spartans constantly expanded their detailed officer hierarchy, allowing for smoother formation evolutions and transmission of orders. Greek armies experimented with the concept of file-closers, who were meant to keep an army from disintegrating from the rear as men were overtaken by terror. They worked ever more closely together with other troop types to increase their versatility and cover their vulnerabilities.

There was experimentation in deployments as well. Early on in the history of the phalanx formation, we already see variations in the number of its ranks as the Greeks tried to work out the best balance between width (a guard against outflanking) and depth (a guard against breakthroughs). As the Classical period wore on, hoplite formations became steadily deeper, as the Greeks worked out that poorly trained men drawn up in deeper formations were more controllable and less likely to break. The very deep phalanx actually allowed the Thebans to defeat the Spartans - despite the latter's advantage in training and capability for manoeuvre - and totally realign the Greek balance of power.

Now, the most spectacular innovation in all of Greek warfare was actually to create something like a hoplite manipular legion, centuries before the Romans conceived of it. The veteran mercenary hoplites of the Ten Thousand, forced to march through inland Anatolia and battle against the hill tribes of the region, grouped their phalanx into blocks of a hundred men so as not to break formation when going uphill, and placed light infantry in the gaps between the blocks in a checkerboard formation. This is exactly the manipular setup of the Roman Republican legion and for exactly the same reason (if the Roman tradition is right that it was developed to fight the Samnites in their mountainous territory). However, when these hoplites made it back to Greece, they never replicated their revolutionary approach. I'll explain why in a moment.

First, we should consider the most important innovation of all. Allegedly it was the Athenian Iphikrates who first reequipped his spear-armed hoplites with pikes, giving them smaller shields so that their hands would be free to wield them. Modern scholars tend to think he did this to fight in Persian service against the Egyptians, who traditionally fought with long pikes. Only some later authors report this, and we don't know how true the story is. However, this Iphikrates later became the godfather of none other than the future king Philip II of Macedon. Philip, when he came to power, reformed the Macedonian army, turning his light infantry levy into professional, well-drilled pikemen - the force that was to topple the Persian Empire and dominate the battlefields of the Mediterranean for centuries to come. Even the Romans were terrified of the Macedonian pike phalanx. The idea came from the Greeks. Eventually, the Greeks themselves adapted to it; many of the city-states encountered by the Romans had upgraded their levies to the use of the pike, if they could afford to train them.

So why do we have this image of Greek military stagnation? As I said, it's important to be aware of when and where the real changes happened. Sparta was better tactically than the other states, with more thorough infantry organisation and drill - but other Greek states refused to adopt the discipline and training needed to achieve this, and therefore never got to explore the possibilities it offered. Only the Ten Thousand ever exceeded the abilities of the Spartans. There was a very simple reason for this: they were professional mercenaries who fought together for a very long, continuous period.

This is the critical point. All other Greek military developments were gradual, incremental, sometimes regressing and going around in circles. But the Ten Thousand, professional, trained, and battle-hardened, had both the experience and the need to develop their tactical skills. The manipular phalanx they developed was a spectacular innovation, but no ordinary untrained Greek levy could ever adopt it. Even the Spartans, who came closest, did not generally have the long service record required to build the trust, the instinctive reactions, and the discipline to make such tactics work.